© 2000 The Operations Research Society of Japan
Journal of the Operations Research
Society of Japan
Vol. 43, No. 1, March 2000
A TWO-PERSON ZERO-SUM GAME WITH FRACTIONAL
LOSS FUNCTION
Yoichi Sawasaki
Yutaka Kimura
Nzzgata University
Kensuke Tanaka
(Received November 25, 1998; Final June 2, 1999)
In this paper, we investigate a two-person zero-sum game with fractional loss function, which
Abstract
we call the two-person zero-sum fractional game. We are interested in a game value and a saddle point of
the two-person zero-sum game and observe that they exist under various conditions. But, in many cases,
it seems to be difficult t o search directly for a game value and a saddle point of the fractional game. Thus,
we study the existence and the properties of a game value and a saddle point for the game with the loss
function including a parameter, which we call the two-person zero-sum parametric game. We show that,
under various conditions, a game value and a saddle point of the parametric game with a special parameter
are those of the fractional game.
1. Introduction
Recently, many concepts and terms about game theory have been introduced and have
been investigated by many authors. Both individual stability and collective stability have
been studied in practical game problems. In view of individual stability in two-person
game, concepts of a game value and a saddle point in the two-person zero-sum game were
introduced. Then, we are interested in a saddle point of the two-person zero-sum game, and
the existence of a saddle point has been actively investigated under various conditions. We
often observe two-person zero-sum games with fractional loss function in many economic
problems. However, as far as we know, we think that there are few papers which treat such
games.
In this paper, we study the existence and some properties of a saddle point for the twoperson zero-sum game with fractional loss function, which we call the two-person zero-sum
fractional game. However, in many cases, it seems to be difficult to search directly for a
saddle point of the fractional game. Because, even if two functions are convex or concave,
the loss function constructing by fractional expression of them is not necessarily convex
or concave. Thus, we consider a two-person zero-sum game with loss function including a
parameter, which we call the two-person zero-sum parametric game. We study the existence
and properties of a saddle point for the parametric game. For the analysis of the parametric
game, the mini-max theorem in convex analysis plays an important role. Then, we show that
under various conditions, a saddle point of the parametric game with a special parameter
is that of the fractional game. Moreover, using a game value of the game with a special
parameter with respect to E > 0, we show that there exists an &-saddlepoint of the fractional
game.
This paper is organized in the following way. In Section 2, we formulate a two-person
zero-sum parametric game and characterize a game value and a saddle point of the parametric game. Section 3 is the main part of this paper. Associated with the results of Section 2,
we discuss relations between the two-person zero-sum parametric game and the two-person
zero-sum fractional game. Moreover, we show that there exists an &-saddle point of the
fractional game.
2. A Two-Person Zero-Sum Parametric Game
We begin by describing a two-person zero-sum parametric game (GPe) by the following
strategic (or normal) form:
(X,y, /,g, 09 Fe),
where
1. X is a subset of a Banach space E, which is called the strategy set of player 1,
2. Y is a subset of a Banach space E, which is called the strategy set of player 2,
3. / : X X Y --+ R and g : X X Y -+ R+, where R+ = (O,oo)?
4. 0 is a real number, which is called a parameter of the game,
5. Fe = /-Og : X X Y -+R, that is, for all (X, Y) 6 X X Y, Fe(x,y) = / ( X , y) -Og(x, y),
is a loss function of player 1 and -Fg ( X , y) is a loss function of player 2.
In general, Fe = infxgx supygyFg(x,y) is called the minimal worst loss of player 1 and
F_Q = supygyinfxgx ^ ( X , y) is called the maximal worst gain of player 2. Further, we call
the duality gap the interval [E@,
Fe].
Definition 2.1
value), if
The two-person zero-sum game (GPe) has a game value ( i n short, a
Fe = L= F,
and this common value F; is called the game value ( i n short, the value) of the game (GPo).
Definition 2.2
A strategy y* E Y is said to be a max-inf of the game (GPe) if
inf Fe ( X , y*) = inf sup Fe(x9y)
xâ‚
xEX yâ‚
and a strategy X* E X is said to be a mini-sup of the game (GPe) if
sup Fe(x*,y) = sup inf Fe(x, Y).
y â‚
Yâ‚
Definition 2.3
game (GPe) if
~â‚
A pair of strategies, (X*, y*) E X
X
Y, is said to be a saddle point of the
inf Fs(x, y*) = Fe(x*,y*) = sup Fe(x*?y ) .
xex
YEY
Then, the following fundamental result is known from J.-P. Aubin [l],Chapter 6.
Proposition 2.1 A pair of strategies, (X*, y*), is a saddle point of the game (GPe) if and
only if X* and y* are a mini-sup and a max-inf of the game (GPe), respectively.
Lemma 2.1 Let X and Y be subsets of a Banach space E and let X be a compact convex
set. Suppose that a function y : X X Y -+ R satisfies the following conditions:
(1) y ( x , y) is lower semi-continuous and convex with respect to X for all y 6 Y ;
(2) y(x, y) is concave with respect t o y for all X E X .
Then, there exists X* E X , which satisfies
sup min p(x, y) = SUP ~ ( x *y), = m i n s u ~Y ( X , Y),
?,â‚ x g x
Y(EY
xEX y g Y
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Game with Fractional Loss Function
that is,
X*
is a mini-sup of the game.
The proof of this lemma is shown by Ky Fan's system theorem. See J.-P. Aubin [l,21,
and K. Fan [5] in detail.
Lemma 2.2 Let X and Y be subsets of a Banach space E and let Y be a compact convex
set. Suppose that a function p : X X Y -+ R satisfies the following conditions:
(1) p(x, y) is upper semi-continuous and concave with respect to y for all X G X ;
(2) p(x, y) is convex with respect to X for all y G Y .
Then, there exists y* G Y , which satisfies
max inf p(x, y) = inf p(x, y*) = inf maxp(x, Y),
yâ‚ xex
xâ‚
xâ‚ Yâ‚
that is, y* is a max-inf of the game.
Corollary 2.1 Let X and Y be compact convex subsets in a Banach space E . Suppose
that a function p : X X Y -+R satisfies the following conditions:
(1) y(x, y) is lower semi-continuous and convex with respect to X for all y G Y ;
(2) y(x, y) is upper semi-continuous and concave with respect to y for all X G X .
Then, there exists a saddle point (X*, y*) G X X Y of the game.
Since there exist a mini-sup and a max-inf of the game under the conditions, the proof
is given by Proposition 2.1.
We have the following minimax theorems and its corollaries for the game (GPe).
Theorem 2.1 Suppose that Y is a compact convex set and that functions f and g satisfy
the following conditions:
(1) f ( X , y) is convex with respect to X for all y G Y ;
(2) f ( X , y) is upper semi-continuous and concave with respect to y for all X G X ;
(3) g(x, y) is concave with respect to X for all y G Y ;
(4) g(x,y) is lower semi-continuous and convex with respect to y for all X C X .
Then, for all 0 0, there exists y* G Y such that
>
-
Fe = & = inf Fe(x,y*),
X
â‚
that is, y* is a max-inf of the game (GPe).
Proof. Since 6 is non-negative, from (2) and (4) in the theorem, it follows that the function
Fe($, y) is upper semi-continuous and concave with respect to y for all X G X. Similarly,
from (1) and (3) in the theorem, we get that the function Fg(x,y) is convex with respect to
X for all y G Y. Thus, Lemma 2.2 shows that Fe= & and that there exists y* G Y which
is a max-inf of the game (GPe).
Corollary 2.2 Suppose that X and Y are compact convex sets and that functions f and
g satisfy the following conditions:
(1) f ( X , y) is lower semi-continuous and convex with respect to X for all y E Y ;
(2) f (X,y ) is upper semi-continuous and concave with respect to y for all X G X ;
(3) g(x, y) is upper semi-continuous and concave with respect to X for all y G Y ;
(4) g(x, y) is lower semi-continuous and convex with respect to y for all X ? X .
Then, for all 0 >.0, there exists a saddle point (X*, y*) G X X Y of the game (GPO),
From Theorem 2.1, there exist a mini-sup X* and a max-inf y* of the game (GPO).Thus,
the proof of the corollary is easily given.
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212
Y. Sa wasaki, Y. Kimura & K. Tanaka
3. A Two-Person Zero-Sum Fractional Game
We define a two-person zero-sum game with fractional loss function ( G P ) as follows:
where X and Y are subsets of a Banach space E, which are called the strategy sets of player
1 and player 2, respectively. Then, using functions f : X X Y -+ R and g : X X Y -+ R+,
a loss function G of player 1 is given by G = f / g , that is, for all (X,y) E X X Y
and -G(x7 y) is a loss function of player 2. Further, two parameters
0 and Q_ are given by
-
0 := inf sup G(x, y) and Q_ := sup inf G(x, y)
x^X y^Y
In general, it holds that
y ^Y
xex
~ < e
-
and the interval [@,,6\ is called "duality gap" of the game (GP).
Definition 3.1
The game ( G P ) has a value i f
Further, y* G Y is said to be a max-inf of the game ( G P ) if
inf G(x, y*) = inf sup G(x ,y) = G.
xex y â ‚
xâ‚
Similarly,
X*
G
X is said to be a mini-sup of the game ( G P ) if
sup G(x*,y) = sup inf G(x, y) = @.
Y^Y
ycY ^X
From now, we will study some relations between the game (GPe) and the game ( G P ) .
Using the parameter G given by (3.3), we consider the game (GP,) with the loss function
such as
F,=f-gg:
XxY-+R,
(3-6)
that is, for each ( x , y ) G X X Y, F7(x,y) = f ( x , y ) - Sg(x,y).
Then, in order to show the main theorems, we observe properties of Fe and .& given
by Fe = infzpx s u p c y Fe(x7y) and /^ = supypyinfxgx Fe(x,y). We shall mention the
following lemmas.
Lemma 3.1 E has the following properties.
(a) Fe is non-increasing with respect to 0.
(b) I ~ F<, 0, it holds that 0 2 9.
(C) If Fe> 0 , it holds that 0 <ç
(d) If 0 > 9,it holds that F, < 0 .
(e) If 0 < 9, it holds that Fe2 0.
Furthermore, under the conditions that Y is compact and that f
continuous with respect to y for all X E X , it holds that
(X,
y) and g(x, y) are
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21 3
Game with Fractional Loss Function
(f) Fe< 0 holds if and only if 0 > 9 holds,
(g) F, 0 holds.
< 02, then we get O1g < Q2gon X X Y, because g is positive on X X Y.
Proof. (a) If
Thus, it follows that f (X, y) - 0^g(x, y) > f (X,y) - O^g(x,y) for all (X,Y) G X X y, that is,
>
Therefore, we get that
-
F = inf sup Fe,(X,y) > inf sup Fe2( X , y) = Fe,.
xEX E Y
x^X yEY
Hence, this shows that Fg is non-increasing with respect to 0.
(b) Since Fe< 0, from the definition of Fe,there exists ?i? 6 X such that
that is, for all y 6 Y,
F@, Y) = f (F,Y) - 0g(x, Y) < 0f(T'y)
This shows that for all y G Y, G(x, y) = -=-g(x, Y)
< 0, that is
sup G@, y) 5 0.
Y â‚
From the definition of 9and (3.9), it follows that 9 $ 8.
(c) Since Fe > 0, that is, for all X G X , supyEyFs(x,y)
depends on X, such that
> 0, there exists yx E Y, which
From (3.10), it follows that G(x,yx) = f ( X, Yx ) > 0, that is, for all
g(x7 Y X )
X
E X,
>
which shows that Q 6.
(d) Since 0 > 0, from the definition of 0, there exists T G X such that
that is, for all y G Y, 0
get that
> G(z,y). This shows that for all y G Y, Fe(?i?,y)< 0. Hence, we
0
2 sup F e ( ~
y ),2 inf sup Fe (X,y) = F,.
xe.x y â ‚
Y â‚
(e) Since 9> 0, from the definition of
9,it follows that for all X G X,
sup G(x, y) > 0.
YEY
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21 4
Y. Sa wasaki, Y. Kimura &K. Tanaka
Thus, there exists y3. E Y, which depends on X, such that G(x, yz) > 0, that is, Fe(x,yx) > 0.
Then, we get that for all X E X ,
Hence, we get
- that
Fe = inf sup Fe(x,y) > 0.
x^X y â ‚
(f) Since F
e < 0, there exists T E X such that supygyF e ( l ,y) < 0, that is, for all
G(1, y) < 0. Noting that Y is compact and G@, y) is continuous on Y, we get that
0 > sup G(Z, y)
Yâ‚
>. inf
sup Gfx, y) = 0.
xEX y â‚
Thus, it holds that 0 > 0.
Conversely, since 6 > 0, from the proof of (b), it follows that there exists T E X such
that for all y E Y, Fe(x, Y) < 0. Noting that Y is compact and Fe(x, Y) is continuous on Y,
we get that
Thus, the proof of (f) is completed.
(g) Suppose that f i < 0 holds, there exists 3 E X such that supyEyFT(%y) < 0. Noting
that Y is compact and Fg{x, y) is continuous on Y, we get that for all y E Y, FT@,y) < 0,
that is, for all y E Y, 9 > G(z, v). This shows that
-
0 > sup G(z, y)
Y â‚
2 inf sup G(x, y) = 9.
xex â
‚
We arrive a t a contradiction. Hence, it is proved that
% > 0 holds.
Lemma 3.2 F_g has the following properties.
is non-increasing with respect to 6 .
(a)
(b) If & < 0 , it holds that 0 6.
(C) I f l e > 0, it holds that 0 0.
(d) If 0 > @,it holds that Eg <0.
(e) If 0 < 0, it holds that Le 0.
Furthermore, under the conditions that X is compact and that f ( X , y ) and g(x, y) are
continuous with respect to X for all y E Y , it holds that
(f) & > 0 holds if and only if 0 < 6.holds,
( g ) &- 5 0 holds.
Proof. Using & and Q_ instead of Feand 0 respectively, we can prove this lemma by similar
arguments to the previous one.
>
<
We have the following relations between the game (GPe) and (GP).
Theorem 3.1 Suppose that y* 6 Y is a max-inf of the game ( G P ) . Then, it holds that
(i) the game ( G P ) has a value Q*,
(ii) If Fg* 5 0, y* is a max-inf of the game (GPe*).
Proof. (i) From the definition of 9 and S., in general it holds that 8 >. ff_.
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21 5
Game with Fractional Loss Function
On the other hand, since y* G Y is a max-infof the game (GP), it follows that
0 = inf G(x, y*) 5 sup inf G(x, Y) = 0.
xex
yeY ^X
Thus, 0 = 9. holds, that is, the game (GP) has a value 0'.
(ii) Since y* is a max-inf of the game (GP), it holds that for all X G X ,
0* = inf G(x7y*)
xcx
that is, for all
X
< G(x, y*),
G X,
Thus, we arrive at the following inequality,
<
0 <; in! F O * ( x , ~ * )inf sup Fs*(x,y)= Fe*5 0.
xcx
xex ycy
This shows that
inf Fe*(X,y*) = inf sup Fe*(X, y).
x e x yâ‚
xcx
That is, y* is a max-inf of the game (GPg*'.
Corollary 3.1
it holds that
Suppose that (X*,y*) 6 X
X
Y is a saddle point of the game (GP). Then,
(i) Fe*(x*,y*)= 0,
(ii) (X*?
y*) is a saddle point of the game (GP0*).
Theorem 3.2 Suppose that the game (GP) has a value 0' and that Fe*
>. 0 holds. Then,
if y* G Y is a max-inf of the game (GP0*),y* is a max-inf of the game (GP).
Proof. Since Fe* 0 and y* is a max-inf of the game (GP0*),it follows that
>
0 5 Fe*= inf sup Fe* (X,y) = inf FP(X,y*)
xex â ‚
xex
which implies that for all
X
< Fo*{x,y*),
for all
X
G X,
GX
Therefore, we get that
0* 5 inf G(x, g*) 5 inf sup G(x, y) = Q*.
xex
x^X yeY
This shows that y* is max-inf of the game (GP).
Corollary 3.2 Suppose that the game (GP) has a value 0' and that a saddle point (X*,y*)
of the game (GP0*)satisfies FO*(x*,y*)= 0 . Then, (X*,y*) is a saddle point of the game
(GP).
Theorem 3.3 Suppose that Y is a compact convex set and that functions f and g satisfy
the following conditions:
1. f (X,y) is convex with respect to X for all y G Y ;
2. f (X,Y) is continuous and concave with respect to y for all X G X ;
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21 6
Y. Sa wasaki, Y. Kimura & K. Tanaka
3. g(x, y) is concave with respect to X for all y G Y ;
4. g(x, y) is continuous and convex with respect to y for all X G X .
Then, if B 0, it holds that
(i) the game ( G P ) has a value Q*,
i i ) There exists y* Y such that y* is a max-inf of the game (GP).
Proof, (i) Since B 0, it follows from Lemma 3.1 (g) and Theorem 2.1 that
>
>
and that there exists y* 6 Y, which is a max-inf of the game (GPF), that is,
5-= sup
inf F,{x,
â‚ x^X
y
y) = inf FF(x, y*).
x6X
From (3.12) and (3.13), it follows that
0
< sup
inf Fg(x, y) = inf Fa(x, y*) < Fa(x, y*),
yâ‚ xcx
xex
From (3.14), it holds that for all
-
Q
X
E X, f
for all
X
E X.
<: G(x, y*). Consequently, we get
inf G(x)Y*)
xex
< sup
inf G(x, y) = Q_.
yâ‚ x^X
This shows that 6 = 0 holds.
(ii) Since Fe*>, 0 and y* E Y is a max-inf of the game (GPe*),from Theorem 3.2, it
follows that y* is a max-inf of the game (GP). Thus, the proof of the theorem is completed.
Theorem 3.4 Suppose that Y is a compact convex set and that functions f and g satisfy
the following conditions:
1. f ( X , y) is convex with respect to X for all y E Y ;
2. f ( X , y) is continuous and concave with respect to y for all X E X ;
3. g(x, y) is concave with respect to X for all y E Y ;
4. g(x, y) is continuous and convex with respect to y for all X E X .
Then, i f 6 00,for any E > 0, there exists a point (X*,Y*) E X X Y such that
>
Q*
<: G(x*,y*) < Q* + E ,
where Q* is the value of the game (GP).
Proof. From Theorem 3.3, we have 9 = S. = Q*. Since Q*
and Theorem 2.1, that there exists y* G Y such that
Fe*=
-
K* = Xinf
Fe*
( X , y*) >. 0.
GX
Then, from (3.15), we get that for all
Q*
On the other hand, since 9< Q*
>. 0, it follows from Lemma 3.1 (g)
X
(3.15)
E X , Fe*
(X,Y*) > 0, that is, for all X 6 X
< G(x, y*).
+ E, it follows from Lemma 3.1 (f) that
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(3.16)
Game with Fractional Loss Function
which shows that there exists X* G X such that s u p g y Fs*+c(x*,y)
for all y G Y, F 6 * + e ( ~ * , <
y )0, that is, for all y E Y,
21 7
< 0. Hence, we get that
Thus, for the particular y* G Y satisfying (3.16), it holds that
Combining (3.16) with (3.17), we show that
O* <, G(x*,y*) < W .
Thus, the proof of the theorem is completed.
Theorem 3.5 Suppose that X and Y are compact convex sets, and that functions f and
g satisfy the following conditions:
1. f ( X , y) is continuous and convex with respect to X for all y G Y ;
2. f ( X , y) is continuous and concave with respect to y for all X G X ;
3. g(x, y) is continuous and concave with respect to X for all y G Y ;
4. g(x, y) is continuous and convex with respect to y for all X E X.
Then, if 0 >. 0, there exists a saddle-point (X*,y*) G X X Y such that for all X and y,
G($*, Y) 5
Q* *X,
y*).
Using Theorem 3.3 and Corollary 2.1, we can easily prove the theorem.
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(North-Holland, Amsterdam, 1982).
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[3] J. P. Crouzeix, J. A. Ferland and S. Schaible: Duality in generalized linear fractional
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[5] K. Fan: Minimax theorems. Proc. Nat. Acad. Set. USA, 39 (1953) 42-47.
[6] R. Jagannathan and S. Schaible: Duality in generalized fractional programming via
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to appear in Annals of Dynamic Game.
[8] Y. Kimura, Y. Sawasaki and K. Tanaka: A non-cooperative equilibrium of n-person
game with fractional loss functions. RIMS Kokyuroku (Kyoto Univ., Kyoto, Japan),
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[g] R.T. Rockafellar: Convex Analysis (Princeton University Press, 1970).
[l01 K. Tanaka and K. Yokoyama: On &-equilibrium point in a noncooperative n-person
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Copyright © by ORSJ. Unauthorized reproduction of this article is prohibited.
Y. Sa wasaki, Y
O Kimura & K. Tanak a
Yoichi Sawasaki and Yutaka Kimura
Department of Mathematics and Information Science
Graduate School of Science and Technology
Niigata University
Niigata 950-2181, Japan
E-mail: sawasakiQqed.sc .niigata-U.ac.J P
yutaka~scux.sc.niigata-u.ac.jp
Kensuke Tanaka
Department of Mathematics
Faculty of Science
Niigata University
Niigata 950-2181, Japan
E-mail: tanakaksQscux.sc .niigata-U.ac .J P
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